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If cos x + cos y =(1)/(3) and sin x +...

If cos x + cos y `=(1)/(3) and sin x + sin y=(1)/(4)` prove that tan `((x+y)/(2))=(3)/(4)`

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To prove that \( \tan\left(\frac{x+y}{2}\right) = \frac{3}{4} \) given that \( \cos x + \cos y = \frac{1}{3} \) and \( \sin x + \sin y = \frac{1}{4} \), we can follow these steps: ### Step 1: Use the sum-to-product identities We can use the sum-to-product identities for cosine and sine: \[ \cos x + \cos y = 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \] \[ ...
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